Professor Erasmus has constructed a special convex pentagon $ABCDE$ that he modestly calls the "Professor-Erasmus-pentagon". The professor claims that he can cut off a smaller pentagon similar to pentagon $ABCDE$ by a straight line.
Question: Has the professor once again made one of his well-known mathematical blunders, or do such pentagons indeed exist?
Note: The five points A,B,C,D,E are all distinct. There are no tricks (like paper folding; pentagons made of rubber; etc).